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Surds and rationalising the denominator
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A surd is an exact root that cannot be simplified to a whole number, like √2. Surds keep answers exact.
Simplifying surds
√(ab) = √a × √b
Look for the largest square factor.
√48 = √16 × √3 = 4√3
Like surds add like algebra: 5√3 + 2√3 = 7√3
Look for the largest square factor.
√48 = √16 × √3 = 4√3
Like surds add like algebra: 5√3 + 2√3 = 7√3
Rationalising a single surd
Multiply top and bottom by the surd on the bottom.
1√5 = √55
6√3 = 6√33 = 2√3
1√5 = √55
6√3 = 6√33 = 2√3
Rationalising a + √b
Multiply top and bottom by the version with the opposite sign. (a + √b)(a − √b) = a2 − b has no surd.
1(2√3 − 1) × (2√3 + 1)(2√3 + 1) = (2√3 + 1)(12 − 1) = (2√3 + 1)11
1(2√3 − 1) × (2√3 + 1)(2√3 + 1) = (2√3 + 1)(12 − 1) = (2√3 + 1)11
Rationalise the denominator of 4(√5 + 1).
- Multiply top and bottom by √5 − 1
- Bottom: (√5 + 1)(√5 − 1) = 5 − 1 = 4
- Top: 4(√5 − 1)
Answer: 4(√5 − 1)4 = √5 − 1
Take out the biggest square factor. Rationalise 1√a by multiplying by √a, and 1(a + √b) by multiplying by a − √b.
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